3 Description

F08JKF (DSTEIN) computes the eigenvectors of a real symmetric tridiagonal matrix T corresponding to specified eigenvalues, by inverse iteration (see Jessup and Ipsen (1992)). It is designed to be used in particular after the specified eigenvalues have been computed by F08JJF (DSTEBZ) with ORDER='B', but may also be used when the eigenvalues have been computed by other routines in Chapters F02 or F08.

If T has been formed by reduction of a full real symmetric matrix A to tridiagonal form, then eigenvectors of T may be transformed to eigenvectors of A by a call to F08FGF (DORMTR) or F08GGF (DOPMTR).

and passes details of the block structure to this routine in the arrays IBLOCK and ISPLIT. This routine can then take advantage of the block structure by performing inverse iteration on each block Ti separately, which is more efficient than using the whole matrix.

On entry: the eigenvalues of the tridiagonal matrix T stored in W1 to Wm, as returned by F08JJF (DSTEBZ) with ORDER='B'. Eigenvalues associated with the first sub-matrix must be supplied first, in nondecreasing order; then those associated with the second sub-matrix, again in nondecreasing order; and so on.

On entry: the first m elements must contain the sub-matrix indices associated with the specified eigenvalues, as returned by F08JJF (DSTEBZ) with ORDER='B'. If the eigenvalues were not computed by F08JJF (DSTEBZ) with ORDER='B', set IBLOCKi to 1, for i=1,2,…,m.

6 Error Indicators and Warnings

INFO<0

If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.

INFO>0

If INFO=i, then i eigenvectors (as indicated by the parameter IFAILV above) each failed to converge in five iterations. The current iterate after five iterations is stored in the corresponding column of Z.

7 Accuracy

Each computed eigenvector zi is the exact eigenvector of a nearby matrix A+Ei, such that

Ei=OεA,

where ε is the machine precision. Hence the residual is small:

Azi-λizi=OεA.

However, a set of eigenvectors computed by this routine may not be orthogonal to so high a degree of accuracy as those computed by F08JEF (DSTEQR).